Wireless power transfer technology has re-emerged as a viable technology for domestic and industrial applications. Recently, multiple-frequency wireless power transmission has been proposed as a means to enhance wireless power transfer. In the article by K. Lee and S. Lukic entitled “Inductive power transfer by means of multiple frequencies in the magnetic link,” IEEE Energy Conversion Congress and Exposition (ECCE), Denver, USA, September 2013, pp: 2912-2919, multi-resonant tanks are used at the transmitter and receiver to amplify and extract power at multiple frequencies. In the Lee article, the power transfer is carried out at a fundamental frequency of 25 kHz and a third harmonic of 75 kHz. Power transfer is spread over more than one frequency so as to increase the power transfer. Single-frequency receivers are used to receiver power sent at different frequencies. For example, if the targeted receiver is tuned at 25 kHz, then transmitting power at 25 kHz will theoretically transfer power to the receiver coil tuned at 25 kHz. The receiver coil tuned at 75 kHz will be the non-targeted receiver and will not receive any power. However, such an approach still has major limitations. A major limitation among them, is that the residual power will be picked up by the non-targeted receiver unless the chosen frequencies are widely separated and/or the quality factors of the resonators are very high (and thus very expensive). The choice of widely separated frequencies also leads to considerable technical and cost constraints on the power transmitter design and the coil resonator design. In the Lee article, the fundamental frequency (25 kHz) and the third harmonic (75 kHz) are added together and the sum of these current components is generated by the power transmitter. However, independent control of the power transfer rates at the two different frequencies cannot be easily controlled.
Some key issues of using multi-frequency for wireless power transfer (WPT) can be understood with reference to the WPT system layout shown in FIG. 1. In FIG. 1 the transmitter T is assumed to be able to operate at more than one frequency. For simplicity, it is assumed that the transmitter can be operated at two operating frequencies, f1 and f2 and that the WPT system has two receivers, A and B, tuned to receive power at frequencies f1 and f2, respectively. In this way, the power flow of each receiver can be controlled separately by controlling the power source with the corresponding frequency.
For simplicity, the relay resonator R is not included in this part of the analysis. The lumped circuit model of the two-receiver system with series compensation is shown in FIG. 2 and its circuit equations are listed below for one operating frequency.(RT+jXT)IT+jωMTAIA+jωMTBIB=VS  (1)jωMTAIT+(RA+jXA)IA+jωMABIB=0  (2)jωMTBIT+jωMABIA+(RB+jXB)IB=0  (3)where ω=2πf; RT is the total resistance in the transmitter loop which includes the source resistance of the source and the parasitic resistance of the inductor and the capacitor; RA and RB are the resistances in the two receiver loops which include the “load” resistance (RLN) (assuming pure resistive load in this analysis) and the “parasitic” resistance (RPN) of the inductors and the capacitors for N=A or B; XN is the reactance in loop N which equals ωLN−1/(ωCN). For example, for Receiver-A, RA=RPA+RLA.
The main power transfer path for each receiver is from the transmitter to the receiver directly. If the receiver is tuned at the operating frequency of the transmitter, the receiver is called the “targeted” receiver. Otherwise, it is called the “non-targeted” receiver. Assume that Receiver-A is tuned to f1, and Receiver-B is tuned to f2. Since Receiver-A and Receiver-B consist of resonators, they are also referred as Resonator-A and Resonator-B.
In the example of FIG. 1, the main power transfer path for frequency f1 is T-A because Receiver-A is the targeted receiver, and the power transfer path T-B-A is traditionally blocked for the power flow of f1 in order to reduce the power reception in Resonator-B at f1. Thus, the example in FIG. 1 can be considered as the combination of two 2-resonator systems. The electrical pioneer Nikola Tesla proved that the resonant frequency of the receiver should equal the operating frequency in order to obtain maximum power transfer efficiency for a 2-resonator system. See the Lee article and U.S. Pat. No. 1,119,732 of N. Tesla entitled “Apparatus for transmitting electrical energy,” which issued Dec. 1, 1914. Therefore, the resonant frequency of Resonator-A should be equal to f1 and the resonant frequency of Resonator-B should be equal to f2. Thus,
                              ω          1                =                                            1                                                                    L                    A                                    ⁢                                      C                    A                                                                        ⁢                                                  ⁢            and            ⁢                                                  ⁢                          ω              2                                =                      1                                                            L                  B                                ⁢                                  C                  B                                                                                        (        4        )            
For each receiver that behaves like a tuned resonator, it operates like a band pass filter. Take Resonator-A as an example. Its current IA(f) can be expressed as:
                                          I            A                    ⁡                      (            f            )                          =                ⁢                                            ω              ⁢                                                          ⁢                              M                TA                            ⁢                              I                T                                                                                      R                  A                  2                                +                                  X                  A                  2                                                              =                    ⁢                                                    ω                ⁢                                                                  ⁢                                  M                  TA                                ⁢                                  I                  T                                                                                                  R                    A                    2                                    +                                                            (                                                                        ω                          ⁢                                                                                                          ⁢                                                      L                            A                                                                          -                                                  1                                                      ω                            ⁢                                                                                                                  ⁢                                                          C                              A                                                                                                                          )                                        2                                                                        =                        ⁢                                                            ω                  ⁢                                                                          ⁢                                      M                    TA                                    ⁢                                      I                    T                                                                                                                                                                ω                          1                          2                                                ⁢                                                  L                          A                          2                                                                                            Q                                                  A                          ⁢                                                                                                          ⁢                          1                                                2                                                              +                                                                  ω                        2                                            ⁢                                                                                                    L                            A                            2                                                    ⁡                                                      (                                                          1                              -                                                                                                ω                                  1                                  2                                                                                                  ω                                  2                                                                                                                      )                                                                          2                                                                                                        =                            ⁢                                                                    M                    TA                                    ⁢                                      I                    T                                                                                        L                    A                                    ⁢                                                                                                              ω                          1                          2                                                                                                      ω                            2                                                    ⁢                                                      Q                                                          A                              ⁢                                                                                                                          ⁢                              1                                                        2                                                                                              +                                                                        (                                                      1                            -                                                                                          ω                                1                                2                                                                                            ω                                2                                                                                                              )                                                2                                                                                                                                                    (        5        )            where ω1=√{square root over (LACA)}; QA1=ω1LA/RA is the Quality Factor of resonator-A at the resonant frequency. FIG. 3 shows the current variations according to the operating frequency f and the quality factor QA1. It is important to note that the shape of the current-frequency characteristic depends on the Quality Factor (or Q-factor) of the coil resonator. A sharp current-frequency characteristic is only possible if the Q-factor is very high (say Q=1000 which is difficult to achieve at low cost). In general, this current-frequency characteristic exhibits a bell-shaped curve with its peak at or near the resonant frequency. Therefore, if the tuned resonant frequency of the non-target receiver is close to that of the target receiver, the non-target receiver will also pick up some current and therefore unintentionally some power. This unintentional power pickup by the non-target receiver is called “cross interference.” On the other hand, if the Q-factor is very high and the current-frequency curve is very sharp, a slight deviation of the operating frequency due to various reasons, such as temperature drift of component values, may cause the power transfer to be reduced drastically.
It is important to note that in the traditional approach the non-targeted receiver resonator is normally not used because it is not the targeted receiver. However, it has been demonstrated that a 3-coil wireless power transfer system (with one relay coil-resonator) can achieve higher energy efficiency than the 2-coil counterpart under some design conditions. See, X. Liu, “Inductive power transfer using a relay coil”, U.S. patent application Ser. No. 13/907,483, filed on 31 May 2013.
From equations (2) and (3), the ratio between the currents in Resonator-A and Resonator-B can be expressed as
                                          I            2                                I            3                          =                                                            M                TA                            ⁢                              R                B                                      +                          j              ⁡                              (                                                                            M                      TA                                        ⁢                                          X                      B                                                        -                                      ω                    ⁢                                                                                  ⁢                                          M                      TB                                        ⁢                                          M                      AB                                                                      )                                                                                        M                TB                            ⁢                              R                A                                      +                          j              ⁡                              (                                                                            M                      TB                                        ⁢                                          X                      A                                                        -                                      ω                    ⁢                                                                                  ⁢                                          M                      TA                                        ⁢                                          M                      AB                                                                      )                                                                        (        6        )            
Because the method allows not only the wireless power transfer at a single frequency to the targeted receiver, but also multiple frequencies to multiple targeted receivers, the following explanations are not restricted to single-frequency operation.
In order to quantify the cross interference introduced by the undesired current in a targeted receiver, an index is proposed that is equal to the ratio of the maximum power caused by the undesired current harmonic and the interested minimum output power generated by the designated current harmonic in the receiver, which is a predetermined value. For example, if the rated output power of a receiver is 5 W, and the untargeted power needs to be limited to within 5% of the targeted power even when the output power is as low as 1/10 of the rated power, then the interested minimum output power of this receiver is 0.5 W. Assuming the general case of transmitting power at both of the frequencies f1 and f2 in FIG. 1, for Resonator-A, the index is
                              δ          A                =                              P                          A              ⁢                                                          ⁢              2              ⁢              max                                            P                          A              ⁢                                                          ⁢              1              ⁢              min                                                          (        7        )            where PA2max is the maximum power caused by the current of f2 and PA1min is the interested minimum power caused by the current of f1 in the equivalent load RA.
Similarly, for Resonator-B,
                              δ          B                =                              P                          B              ⁢                                                          ⁢              1              ⁢              max                                            P                          B              ⁢                                                          ⁢              2              ⁢              min                                                          (        8        )            where PB1max is the maximum power caused by the current of f1 and PB2min is the interested minimum power caused by the current of f2 in the equivalent load RB. It can be seen from equations (7) and (8) that a large index implies that the cross interference is significant and the situation is not desirable.
By utilizing equations (6), (7) and (8) the index can be re-written as
                              δ          A                =                                                            P                                  B                  ⁢                                                                          ⁢                  2                  ⁢                  max                                            ⁢                              R                LA                                                                    P                                  A                  ⁢                                                                          ⁢                  1                  ⁢                  min                                            ⁢                              R                LB                                              ·                                                                      M                  TA                  2                                ⁢                                  R                  B                  2                                            +                                                ω                  2                  2                                ⁢                                  M                  TB                  2                                ⁢                                  M                  AB                  2                                                                                                      M                  TB                  2                                ⁢                                  R                  A                  2                                            +                                                (                                                                                    M                        TB                                            ⁢                                                                        L                          A                                                ⁡                                                  (                                                                                    ω                              2                                                        -                                                                                          ω                                1                                2                                                            /                                                              ω                                2                                                                                                              )                                                                                      -                                                                  ω                        2                                            ⁢                                              M                        TA                                            ⁢                                              M                        AB                                                                              )                                2                                                                        (        9        )                                          δ          B                =                                                            P                                  A                  ⁢                                                                          ⁢                  1                  ⁢                  max                                            ⁢                              R                LB                                                                    P                                  B                  ⁢                                                                          ⁢                  2                  ⁢                  min                                            ⁢                              R                LA                                              ·                                                                      M                  TB                  2                                ⁢                                  R                  A                  2                                            +                                                ω                  1                  2                                ⁢                                  M                  TA                  2                                ⁢                                  M                  AB                  2                                                                                                      M                  TA                  2                                ⁢                                  R                  B                  2                                            +                                                (                                                                                    M                        TA                                            ⁢                                                                        L                          B                                                ⁡                                                  (                                                                                    ω                              1                                                        -                                                                                          ω                                2                                2                                                            /                                                              ω                                1                                                                                                              )                                                                                      -                                                                  ω                        1                                            ⁢                                              M                        TB                                            ⁢                                              M                        AB                                                                              )                                2                                                                        (        10        )            
A design example is shown in connection with the system of FIG. 4, wherein the given parameters are: LA=LB=81.3 μH; RPA=RPB=0.85Ω; MTA=MAB=2.6624 μH; MTB=0.49 μH; δA=10%; δB=10%; PA1max=PB2max=2.5 W; PA1min=PB2min=0.25 W; f1=600 kHz; f2=500 kHz. The calculated values for the load resistance by solving equations (9) and (10) are RLA=1.49Ω and RLB=1.24Ω. The load resistance values are small in order to increase the quality factors of the receivers according to the previous analysis. However, the small load resistance values might lead to low efficiency. In this case, the overall efficiency is 48.2% while the possible maximum efficiency of the system is 59.6% if the load resistance values are optimized, which are RLA=10.06Ω and RLB1.76Ω. The load resistance and the operating frequencies could be further adjusted to obtain higher power transfer efficiency, but there are always compromises to make between the frequency difference and the efficiency (decided by operating frequencies and load resistance values). Also, it should be noted that equations (9) and (10) are only valid for a narrow frequency range in which the AC resistance of the resonator can be considered as constant.
In addition to the difficulty of achieving high efficiency, the drawbacks of the traditional method also include that: (1) it cannot remove the undesired current substantially; (2) the indirect power transfer paths (for example T-A-B for Resonator-B in the system in FIG. 4) are not utilized, which is a waste of the power transfer capability of the system; and (3) the interferences are highly sensitive to the resonant frequencies of the resonators (i.e. the inductance and capacitance values of the resonator) due to the high quality factors.
There are many practical applications in which indirect power paths should be utilized in order to raise the power transfer capability. For example, for the system shown in FIG. 4, the direct path for Resonator-B is T-B and the indirect power path is T-A-B. The power transfer efficiency of the system will be much lower if only the direct path is used. It has been demonstrated by the inventors that the cross-coupling (or indirect) power transfer paths can be utilized to further increase the capacity of power transfer at a single frequency for a single load. See, C. K. Lee, W. X. Zhong and S. Y. R. Hui, “Effects of Magnetic Coupling of Non-adjacent Resonators on Wireless Power Domino-Resonator Systems”, IEEE Trans. Power Electronics, vol. 27, no. 4, pp. 1905-1916, April 2012